Tilburg University
On the spectral characterization of mixed extensions of P-3
Haemers, Willem H.; Sorgun, Sezer; Topcu, Hatice
Published in:
The Electronic Journal of Combinatorics: EJC
Publication date:
2019
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Citation for published version (APA):
Haemers, W. H., Sorgun, S., & Topcu, H. (2019). On the spectral characterization of mixed extensions of P-3. The Electronic Journal of Combinatorics: EJC, 26(3), [P3.16].
https://www.combinatorics.org/ojs/index.php/eljc/article/view/v26i3p16
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On the spectral characterization
of mixed extensions of P
3Willem H. Haemers
Dept. of Econometrics and O.R., Tilburg University, Tilburg, The Netherlands
haemers@uvt.nl
Sezer Sorgun
Hatice Topcu
Dept. of Mathematics,
Nev¸sehir Hacı Bekta¸s Veli University, Nev¸sehir, Turkey srgnrzs,haticekamittopcu@gmail.com
Submitted: Oct 31, 2018; Accepted: Jun 29, 2019; Published: Jul 19, 2019 c
The authors. Released under the CC BY-ND license (International 4.0).
Abstract
A mixed extension of a graph G is a graph H obtained from G by replacing each vertex of G by a clique or a coclique, whilst two vertices in H corresponding to distinct vertices x and y of G are adjacent whenever x and y are adjacent in G. If G is the path P3, then H has at most three adjacency eigenvalues unequal to 0 and
−1. Recently, the first author classified the graphs with the mentioned eigenvalue property. Using this classification we investigate mixed extension of P3 on being
determined by the adjacency spectrum. We present several cospectral families, and with the help of a computer we find all graphs on at most 25 vertices that are cospectral with a mixed extension of P3.
Mathematics Subject Classifications: 05C50
1
Introduction
Here this classification is applied to mixed extensions of the path P3 (see next section),
which have the mentioned spectral property. We give the characteristic polynomial of all graphs in the classification, and completely determine the mixed extensions of P3 which
are determined by the spectrum on at most 25 vertices. Also we present several infinite families of cospectral graphs with all but three eigenvalues equal to 0 or −1.
2
Mixed extensions of P
3Let G be a graph with vertex set V (G) = {1, . . . , m} and let V1, . . . , Vm be mutually
disjoint nonempty finite sets. A graph H with vertex set V (H) = V1∪ . . . ∪Vm is defined
as follows. For each i ∈ {1, . . . , m}, all vertices of Vi are either mutually adjacent (form
a clique), or mutually nonadjacent (form a coclique). For any u ∈ Vi and v ∈ Vj (i 6= j)
{u, v} in an edge in H if and only if {i, j} is an edge in G. The graph H is called a mixed extension of G. A mixed extension is represented by an m-tuple (t1, . . . , tm) of nonzero
integers, where ti > 0 indicates that Vi is a clique of order ti and ti < 0 means that
Vi is a coclique of order −ti. A mixed extension of G is a special case of a generalized
composition or G-join, introduced in [3] and [6], respectively. We refer to [5], [3] and [6] for basic results on mixed extensions, and to [1] or [4] for graph spectra.
Suppose H is a mixed extension of the path P3 of type (t1, t2, t3). Then the adjacency
matrix of H admits the following structure. (As usual, J is an all-ones matrix, Jn is the
n × n all-ones matrix, and In is the identity matrix of order n.)
A = ε1(J|t1|− I|t1|) J O J ε2(J|t2|− I|t2|) J O J ε3(J|t3|− I|t3|) ,
where εi = 1 if ti > 1 and εi = 0 otherwise (i = 1, 2, 3).
The given partition of A is equitable, therefore A has two kinds of eigenvalues: the ones that have eigenvectors in the span V of the characteristic vectors of the partition, and those whose eigenvectors are orthogonal to V . The first kind coincide with the eigenvalues of the quotient matrix
Q = ε1(|t1| − 1) |t2| 0 |t1| ε2(|t2| − 1) |t3| 0 |t2| ε3(|t3| − 1) .
The second kind of eigenvalues of H remain eigenvalues if we subtract an all-one block from each nonzero block of A. So these eigenvalues are also eigenvalues of
B = −ε1I|t1| O O O −ε2I|t2| O O O −ε3I|t3| ,
which are clearly all equal to 0 or −1. So, if n = |t1| + |t2| + |t3| is the order of H, and
polynomial of A equals p(x) = q(x)(x+1)bxn−b−3. Indeed, the exponent of (x+1) equals b,
because the coefficient of xn−1 in p(x) equals trace(A) = 0. Note that d = − det(Q) > 0,
and det(Q) = 0 if and only if ε1 = ε3 = 0 that is, both end vertices of P3 are replaced by
a coclique. If this is the case then we call it an improper mixed extension of P3, because
it is in fact a mixed extension of P2. If det(Q) 6= 0, then the mixed extension is proper.
Proposition 1. A proper mixed extension of the path P3 has exactly two positive
eigen-values and one eigenvalue smaller than −1.
Proof. The quotient matrix Q has at least one positive eigenvalue, and det(Q) < 0 gives that Q (and A) has exactly two positive eigenvalues. It is well-known that a graph with smallest adjacency eigenvalue at least −1 is the disjoint union of complete graphs. Therefore the smallest eigenvalue of A (and Q) is less than −1.
3
Spectral characterizations
Some known results on spectral characterizations of graphs deal with special cases of mixed extensions of P3. This includes the pineapple graphs (type (p, 1, −q), p, q > 0), and
the complete graphs from which the edges of a complete bipartite subgraph are deleted (type (p, q, r), p, q, r > 0). The latter graphs are determined by their spectrum for all p, q, r > 0; see [2]. For the pineapple graphs it is known for which p and q the graphs are determined by its spectrum. For the precise conditions on p and q we refer to [8]. It follows that among the connected graphs the pineapple graph is determined by its spectrum.
An improper mixed extension of P3 is either a complete bipartite graph Kp,q (p >
2, q > 1), or a complete split graph CSp,q (p, q > 2), which is a complete graph Kp+q from
which the edges of a complete subgraph Kq are deleted. For these classes of graphs the
spectral characterization is straightforward and known. The complete split graph CSp,q
is determined by its spectrum for all p, q > 2. The complete bipartite graph Kp,q is not
determined by its adjacency spectrum if and only if pq has a divisor r strictly between p and q. Then Kr,pq/r extended with p + q − r − pq/r isolated vertices is cospectral with
Kp,q. From now on we restrict to proper mixed extensions of P3. We define G to be the
set of graphs with all but at most three adjacency eigenvalues equal to −1 or 0. Note that a graph G in G remains in G if isolated vertices are added or deleted. Therefore, for the classification, we may restrict to the set of graphs in G with no isolated vertices. By G00 we denote the class of graphs in G with no isolated vertices, with two positive eigenvalues and with one eigenvalue less than −1. By Proposition 1 we know that a proper mixed extension of P3 is in G00. The characterization of graphs in G, mentioned in
the introduction, when restricted to G00, is given by the following two theorems.
Theorem 2. A disconnected graph H belongs to G00, if and only if H is one of the following.
(i) Kp+ Kq,r with p, q, r > 2,
b c d (−p, −q, r) r − 1 pq + qr pq(r − 1) (−p, q, r) q + r − 2 q + r + pq − 1 pq(r − 1) (p, −q, r) p + r − 2 q(p + r) + (p − 1)(1 − r) qr(p − 1) + pq(r − 1) (p, q, r) p + q + r − 3 2q + 2r + 2p − pr − 3 qpr + pr − p − q − r + 1 (−2, q, r, −2) q + r − 1 2q + 2r 4qr (−3, q, −2, s) q + s − 1 2s + 5q − qs 6qs (−2, −2, −3, s) s 2s + 10 12s (5, 2, −r, 4) 8 6r − 9 34r + 18 (4, 2, −r, 6) 9 8r − 15 40r + 25 (7, 2, −r, 3) 9 5r − 6 37r + 16 (3, 3, −r, 6) 9 9r − 15 45r + 25 (4, 3, −r, 3) 7 6r − 4 30r + 12 (7, 3, −r, 2) 9 5r + 1 37r + 9 (3, 4, −r, 4) 8 8r − 9 40r + 18 (3, 6, −r, 3) 9 9r − 6 45r + 16 (4, 6, −r, 2) 9 8r + 1 40r + 9 (5, 4, −r, 2) 8 6r + 1 34r + 8 (2, 2, 2, 7) 9 1 65 (2, 2, 6, 3) 9 9 73 (2, 2, 3, 4) 7 5 51 (2, 3, 2, 5) 8 1 68 (2, 3, 4, 3) 8 7 74 (2, 5, 2, 4) 9 1 89 (3, 2, 2, 3) 6 3 40 (2, 5, 3, 3) 9 6 94 (1, p, −q, r, 1) p + r − 1 (q + 1)(p + r) − pr pr(2q + 1) Kp+ Kq,r p − 1 qr qr(p − 1) Kp+ CSq,r p + q − 2 qr − (p − 1)(q − 1) qr(p − 1)
Theorem 3. A connected graph H belongs to G00 if and only if H is one of the following. (i) A mixed extension of P3 of type (−p, −q, r); (−p, q, r); (p, −q, r), or (p, q, r) with
p, q > 1 and r > 2,
(ii) a mixed extension of P4 of type (−2, q, r, −2); (−3, q, −2, s), or (−2, −2, −3, s) with
q, r, s > 1,
(iii) a mixed extension of P4 of type (p, q, −r, s) with r > 1 and (p, q, s) ∈ (5, 2, 4),
(4, 2, 6), (7, 2, 3), (3, 3, 6), (4, 3, 3), (7, 3, 2), (3, 4, 4), (3, 6, 3), (4, 6, 2), (5, 4, 2) ,
(iv) a mixed extension of P4 of type (p, q, r, s), with (p, q, r, s) ∈ (2, 2, 2, 7), (2, 2, 6, 3),
(2, 2, 3, 4), (2, 3, 2, 5), (2, 3, 4, 3), (2, 5, 2, 4), (2, 5, 3, 3), (3, 2, 2, 3) , (v) a mixed extension of P5 of type (1, p, −q, r, 1) with p, q, r > 1.
The characteristic polynomial p(x) of a graph in G00 with n vertices can be written as: p(x) = (x3− bx2− cx + d)(x + 1)bxn−b−3
, with b > 0 and d > 0
(note that the multiplicity of the eigenvalue −1 equals b, because trace(A) = 0). So the nonzero part of the spectrum of a graph in G00 is determined by the coefficients b, c and d. Table 1 gives these coefficients for each type of the above classification. If two graphs in the classification have the same b,c and d, then the nonzero part of the spectrum is the same. We will call such a pair pseudo-cospectral. If two pseudo-cospectral graphs have different order, we can extend the smaller graph (or both graphs) with some isolated vertices, so that the two graphs become cospectral. Therefore Table 1 gives all information needed to decide which proper mixed extensions of P3 are determined by their spectrum.
Nevertheless, it is far too complicated to give a general result. Therefore we present some special cases.
Theorem 4. [2] Suppose H is a mixed extension of P3 of order n and type (p, q, r) with
p, q, r > 1. Then H is determined by the spectrum of its adjacency matrix.
Proof. From Table 1 we see that for this case the coefficient b equals p + q + r − 3 = n − 3. For every graph H0 of order n0 pseudo-cospectral with H, which belongs to one of the other types, we have b < n0− 3. So H0 has more vertices than H, and we cannot obtain a
graph cospectral with H by adding isolated vertices to H0. If H0 is a mixed extension of P3 of type (p0, q0, r0) with p0, q0, r0 > 1, and H0 is cospectral with H, then H and H0 have
the same coefficients b, c an d, which implies p = p0, q = q0, and r = r0. The following results follow straightforwardly from Table 1.
Proposition 5. The following types of mixed extensions of P3 are pseudo-cospectral with
a non-isomorphic graph in G00.
(ii) type (p, (p − 1)(p − 2), p) with Kp(p−1)+ Kp−1,p−1, where p > 3,
(iii) type (p, q, p) with Kp + CSp+q−1,r where r = 1 + pq/(p + q − 1) is an integer, and
p > 2, q > 1.
Notice that this proposition does not give any graph which is cospectral and non-isomorphic with a mixed extension of P3, because in all three cases the second graph has
more vertices than the first one. For the pineapple graph Kpq, which is a mixed extension of P3 of type (p − 1, 1, −q), we find (see [7]).
Proposition 6. The pineapple graph K2pp2 is cospectral with the mixed extension of P3 of
type (p, −p, p) extended with p(p − 1) isolated vertices.
See [8] for more examples of graphs non-isomorphic but cospectral with Kq
p. As
re-marked before, there exists no connected example. However, there do exist connected non-isomorphic cospectral mixed extensions of P3.
Proposition 7. The mixed extensions of P3 of types (p, −q, q(2q − p − 1)/(q − p)) and
(−q, 2q − 1, p(2q − p − 1)/(q − p)) are cospectral whenever q(2q − p − 1)/(q − p) is a positive integer, and p, q > 1.
Note that there are infinitely many values of p and q for which the above fraction is integral. For example if p = 1 or p = q − 1.
4
Enumeration
Using Table 1, we generated by computer (using Maple) a list of all non-isomorphic graphs in G00 on at most 25 vertices. The list contains almost 10000 graphs. We ordered the graphs lexicographically with respect to the coefficients b, c, d. Then the pseudo-cospectral graphs in G00become consecutive items in the table with the same b, c, d. Since the list is very long we only consider the mixed extensions of P3 that have at least one
non-isomorphic pseudo-cospectral mate in the list. By use of the shortened list we found the pseudo-cospectral examples of Propositions 5 and 7. Next we deleted the cases given in Proposition 5, 6 and 7 from the list. The final list is given in Table 2. Thus this table together with Propositions 5 to 7 give all mixed extensions of P3 of order n 6 25 for
which there exist at least one pseudo-cospectral graph in G00 of order at most 25. Since Proposition 5 gives no graphs cospectral with a mixed extension of P3, we can conclude
the following:
Theorem 8. Suppose H is a proper mixed extension of P3 of order n 6 25. Then H
Acknowledgments
We thank one referee for the correction of some nasty errors, and another referee for pointing out the references [3] and [6]. This work is partially supported by TUBITAK (the Scientific and Technological Research Council of Turkey) research project 117F489.
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